Quaternionic vector space

In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H is the (non-commutative) division ring of quaternions. The space Hⁿ of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:

[math]\displaystyle{ q (q_1,q_2,\ldots q_n) = (q q_1,q q_2,\ldots q q_n) }[/math]

[math]\displaystyle{ (q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q) }[/math]

for quaternions q and q₁, q₂, ... q_n.

Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hⁿ for some n.

See also

• Vector space
• General linear group
• Special linear group
• SL(n,H)
• Symplectic group

References

• Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.